Let $R=\mathbf{k}[X_1,\ldots,X_n]$.
Definition: For $f\in R$, let $\mathcal{J}_f$ denotes the ideal generated by the partial derivatives of $f$ (Jacobian ideal), namely $$\mathcal{J}_f = \left\langle\frac{\partial f}{\partial X_1},\ldots,\frac{\partial f}{\partial X_n} \right\rangle$$
Question: Does there exist $f\in R$ such that $f^m\in \mathcal{J}_f$ for some $m\in\mathbb{N}$, but $f\not\in\mathcal{J}_f$?
In other words is it possible that $\mathcal{J}_f\subsetneq \sqrt{\mathcal{J}_f}$?